Dividends: From Refracting to Ratcheting

In this paper we consider an alternative dividend payment strategy in risk theory, where the dividend rate can never decrease. This addresses a concern that has often been raised in connection with the practical relevance of optimal classical dividend payment strategies of barrier and threshold type. We study the case where once during the lifetime of the risk process the dividend rate can be increased and derive corresponding formulae for the resulting expected discounted dividend payments until ruin. We first consider a general spectrally-negative Lévy risk model, and then re fine the analysis for a diffusion approximation and a compound Poisson risk model. It is shown that for the diffusion approximation the optimal barrier for the ratcheting strategy is characterized by an unexpected relation to the case of refracted dividend payments. Finally, numerical illustrations for the diffusion case indicate that with such a simple ratcheting dividend strategy the expected value of discounted dividends can already get quite close to the respective value of the refracted dividend strategy, the latter being known to be optimal among all admissible dividend strategies.